Abstract Algebra: Suitable for Self-Study or Online Lectures (Mathematics Study Resources, 7)

Preface
Contents
1 Motivation and Prerequisites
	1.1	Goals
		1.1.1	Algebraic Structures
		1.1.2	Polynomial Equations in One Variable
	1.2	Prerequisites
2 Field Extensions and Algebraic Elements
	2.1	Field Extensions
	2.2	Intermediate Fields and Algebraic Elements
3 Groups
	3.1	General Definition and Consequences
	3.2	Subgroups and Group Homomorphisms
4 Group Quotients and Normal Subgroups
	4.1	Equivalence Relations
	4.2	Group Quotients
	4.3	The Lagrange Theorem
	4.4	Normal Subgroups and Factor Groups
	4.5	The Fundamental Homomorphism Theorem for Groups
	4.6	Finite Cyclic Groups
5 Rings and Ideals
	5.1	Commutative Rings with Unity
	5.2	Ring Homomorphisms
	5.3	Units and Zero Divisors
	5.4	Ideals, Factor Rings, and the Fundamental Homomorphism Theorem
	5.5	Prime Ideals and Maximal Ideals
	5.6	The Chinese Remainder Theorem
	5.7	Examples of Rings in Quadratic Number Fields
6 Euclidean Rings, Principal Ideal Rings, Noetherian Rings
	6.1	Euclidean Rings
	6.2	The Euclidean Algorithm
	6.3	Noetherian Rings
7 Unique Factorization Domains
	7.1	Prime Elements and Irreducible Elements, Unique Factorization Domains
	7.2	Properties
8 Quotient Fields for Domains
9 Irreducible Polynomials in UFDs
	9.1	Content of Polynomials
	9.2	Reduction Modulo Prime Elements
	9.3	The Gauss Lemma
	9.4	Application of the Reduction Mod p
10 Galois Theory (I)—Theorem A and Its Variant A’
	10.1	The Miraculous Creation of some Field
	10.2	The Splitting Field
	10.3	Theorem A and A’
	10.4	Application for the Tower of Field Extensions
	10.5	The Galois Group
11 Intermezzo: An Explicit Example X5 – 777X + 7
12 Normal Field Extensions
	12.1	Algebraic Closure
	12.2	Extension of Field Homomorphisms
	12.3	Normal Extensions
13 Separability
	13.1	Motivation and Definition
	13.2	Formal Derivation
	13.3	Characteristic of a Field and Separability
	13.4	The Degree of Separability
	13.5	The Primitive Element Theorem
14 Galois Theory (II)—The Fundamental Theorem
	14.1	The Fundamental Theorem of Galois Theory—Statement
	14.2	Outlook on an Application—Quadratic Formula for All Degrees?
	14.3	Proof of the Fundamental Theorem
	14.4	Proof of the Addendum
15 Cyclotomic Fields
	15.1	Roots of Unity
	15.2	Cyclotomic Fields and Cyclotomic Polynomials
16 Finite Fields
	16.1	Prime Fields, Finite Fields, and Frobenius
	16.2	Finite Fields
17 More Group Theory—Group Actions and Sylow\'s Theorems
	17.1	Group Actions
	17.2	The Sylow Theorems
	17.3	Applications of Sylow’s Theorems and Some Common Tricks
	17.4	Proof of Sylow’s Theorems
18 Solvability of Polynomial Equations
	18.1	Solvable Groups
	18.2	Solving Polynomial Equations by Radicals
	18.3	The General Equation of Degree n
Proof of the Existence of an Algebraic Closure
Tricks and Methods to Classify Groups of a Given Order